Crossing Point

Where the two lobes of L meet (algebraic version). The crossing point is where B-sector and C-sector intersect; not a smooth point. Geometric S^1 v S^1 emerges in Book II.

Crossing Point

Summary

Where the two lobes of L meet (algebraic version). The crossing point is where B-sector and C-sector intersect; not a smooth point. Geometric S^1 v S^1 emerges in Book II.

Statement

%
\label{def:crossing-point}
The \textbf{crossing point} of the algebraic lemniscate
$\mathbb{L} = \hat{\mathbb{Z}}_\tau[j]$
is the element:
\[
    \boxed{%
    \omega_\times := 1 = e_+ + e_-.}
\]
It is the unique element of $\mathbb{L}$
whose spectral coefficients are both equal:
$\chi_+(\omega_\times) = \chi_-(\omega_\times) = 1$.
Equivalently, $\omega_\times$ is the multiplicative identity
of the ring $\mathbb{L}$,
and it is the unique element
that belongs to \textbf{both} sectors simultaneously:
$\omega_\times \cdot e_+ = e_+ \neq 0$
and $\omega_\times \cdot e_- = e_- \neq 0$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 90
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part11/ch45-crossing-point.tex lines 64-83

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Boundary.Fourier
• Name: Tau.Boundary.crossing_point

Dependencies

• Canonical: I.D37, I.T12, I.D18

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.